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CERTAIN CURVATURE PROPERTIES OF LP-SASAKIAN MANIFOLD WITH RESPECT TO QUARTER SYMMETRIC NON METRIC CONNECTION
Department of Mathematics & AstronomyUniversity of Lucknow, Lucknow, 226007 UP INDIA
n.v.c.shukla , ram asray verma
Abstract- The object of the present paper is to study of various curvature tensor on an Lorentzian para-Sasakian manifold with respect to quarter-symmetric non-metri connection.
Mathematics Subject Classification- [2010]53C07, 53C25
Key words and phrases- Lorentzian para-Sasakian manifold, Projective tensor, Psudo Projective tensor,Concircular curvature tensor,Quasi-Concircular curvature tensor
1 Introduction
In 1975,Golab [4] introduced the notion of quartersymmetric connection in a Riemannian manifold with affine connection. This was further developed by Yano and Imai [27], Rastogi [22], Mishra and Pandey [10], Mukhopadhyay, Roy and Barua [11], Biswas and De [3], Sengupta and Biswas [23], Singh and Pandey [25] and many other geometers.In this paper we define and study a quartersymmetric nonmetric connection on an Lorentzian paraSasakian manifold . Section 1 , is introductory . In section 2 , we give the definition of Lorentzian paraSasakian . In section 3 , we define a quartersymmetric non
metric connection . In section 4 , curvature tensor Ricci tensor and scalar curvature tensor of the quartersymmetric nonmetric connection is obtained . In section 5 , we have A necessary and sufficient condition for the Ricci tensor of to be symmetric and skew symmetric . In section 6 ,The Projective curvature tensor of the quartersymmetric non metric connection in Lorentzian paraSasakian manifold is studied. In section 7 ,We have studied Psudo Projective curvature tensor of the quartersymmetric nonmetric connection in Lorentzian paraSasakian manifold is studied . In section 8,The Concircular curvature tensor of the quartersymmetric nonmetric connection in Lorentzian para-sasakian and finally In section 9,The QuasiConcircular curvature tensor of the quartersymmetric non
2 Preliminaries
An ndimensional differential manifold n
M is a Lorentzian paraSasakian (LP Sasakian) manifold if it admits a (1,1)tensor field , contravariant vector field ,a covariant vector field , and a Lorentzian metric g, which satisfy
, ) ( =
2
X X X (2.1)
1, = ) (
(2.2)
), ( ) ( ) , ( = ) ,
( X Y g X Y X Y
g (2.3)
), ( = ) ,
(X X
g (2.4)
, ) ( ) ( 2 ) ( ) , ( = ) )(
(X Y g X Y Y X X Y (2.5)
, = X
X
(2.6)
for any vector fields X and Y, where denotes covariant differentiation with respect to metric g ([7], [14] and [8]).
In an LPSasakian manifold Mn with structure (,,, g), it is easily seen that
1), ( = ) ( ,
0 = ) ( , 0
= X rank n
(2.7)
). , ( = ) ,
(X Y g X Y
F (2.8)
Then the tensor field F is symmetric (0,2) tensor field
), , ( = ) ,
(X Y F Y X
F (2.9)
). )( ( = ) ,
(X Y Y
F X (2.10)
3 The Quartersymmetric nonmetric Connection in a LPSasakian manifold
Let
Mn,g
be an LPSasakian manifold with LeviCivita connection we definelinear connection on Mn by
,
, = Y X Y X Y Y X g X Y Y X
X
(3.1)
for all vector fields X and Y(TMn).
. ) ( )
( = ) ,
(X Y Y X X Y
T (3.2)
A linear connection satisfying
3.2 is called quartersymmetric connections ([5],[13],[17],[20]).Further
using
3.1 we have
Xg
Y,Z
=2
X gY,Z
2
X gY,Z
. (3.3) A linear connection defined by
3.1 satisfies
3.2 and
3.3 is called quartersymmetric nonmetric connection.Let be a linear connection in Mn given by
,
. = Y H X YY X
X
(3.4)
Now we shall determine the tensor field H such that satisfies
3.2 and
3.3 From
3.4 we have ,
X,Y
=H
X,Y
H
Y,X
.T (3.5)
We have
X,Y,Z
=
g
Y,Z
.G X (3.6)
From
3.4 and
3.6 we have
H X,Y ,Z
g
H
X,Z
,Y
= G
X,Y,Z
.g (3.7)
From
3.4,
3.6,
3.7 and
3.3 we have ,
T X Y Z
g
T
Z X
Y
g
T
Z Y
X
g
H
X Y
Z
g
H
Y X
Z
g , , , , , , = , , , ,
H Z X Y
g
H
X Z
Y
g , , , ,
H Z Y X
g
H
Y Z
X
g , , , ,
H X Y Z
G X Y Z
g , , , ,
=
Y X Z
G Z X Y
G , , , ,
H X Y Z
g , ,
2 =
X g Y,Z
2
X g Y,Z
2
Y g X,Z
2
Y g X,Z
2
,
2
,
,2 Z g X Y Z gX Y
X Y
T
X Y
T
X Y
T
Y X
H , , ,
2 1 =
, (3.8)
(X)Y (Y)X g(X,Y)
. ) , ( )
( )
(
X Y Y X g X Y
Where T be a tensor field of type
1,2 defined by
T X,Y ,Z
= g
T
Z,X
,Y
.g (3.9)
From
3.2 and
3.9 we have ,
X,Y
=(X)Y g(X,Y).Using
3.2 and
3.10 in
3.8 we get ,4 Curvature tensor of an LPSasakian manifold with respect to the Quarter symmetric nonmetric Connection
Let R and R be the curvature tensor of the connection and respectively
is the curvature tensor of with respect to the Riemannian Connection . Contracting
4.3 , we get
Y Z
SY Z
n
g Y Z
S , = , 3 , (4.4)
1
1
2
, 2=r n n n
r (4.5)
where S and r are the Ricci tensor and scalar curvature with respect to and trace
=
.
Theorem 2 The Curvature tensor R(X,Y)Z and Ricci tensor S
Y,Z
and scalar curvature r of an LPSasakian manifold with respect to quartersymmetric nonmetric connection as given by
4.3, 4.34.4
and
4.5 respectively.Let us assume that R(X,Y)Z=0 in
4.3 we get ,
Y Z
n
g Y Z
n
g Y Z
S , = 3 , 3 ,
2
Y Z , which gives
1
1
2
2
= n n n
r
Theorem 3 If an LPSasakian manifold M admits a quartern symmetric non metric connection whose curvature tensor vanishes, then the scalar curvature r is given by
1
1
2
.2
= n n n
r
From
4.3 we get ,
X,Y,Z,W
RY,X,Z,W
=0.R (4.6)
Again by virtue of
4.3 and first Binachi’s identity with respect to connection we have
X,Y,Z,W
RY,Z,X,W
R Z,X,Y,W
=0.R
5 Projective Curvature tensor on an LPSasakian manifold with respect to the Quartersymmetric nonmetric Connection
Let P and P denote the projective curvature tensor with respect to connection and respectively. then
1
( , ) ( , )
,1 )
, ( = ) ,
( S Y Z X S X Z Y
n Z Y X R Z Y X
P
(5.1)
and
1
( , ) ( , )
.1 )
, ( = ) ,
( S Y Z X S X Z Y
n Z Y X R Z Y X
P
(5.2)
Y
Hence, we can state the following theorem;
Theorem 4 In an LPSasakian manifold Projective Curvature tensor with respect to Riemannian connection is equal to the Projective Curvature tensor with respect to Quartersymmetric nonmetric Connection if equation
5.4 holds.6 Psudo Projective Curvature tensor on an LPSasakian manifold with respect to the Quartersymmetric nonmetric Connection
n
S Y Z X S X Z Y
Hence, we can state the following theorem;
Theorem 5 In an LPSasakian manifold Psudo Projective Curvature tensor with respect to Riemannian connection is equal to the Psudo Projective Curvature tensor with respect to Quartersymmetric nonmetric Connection if equation
6.4 holds.Let C and C denote the Concircular curvature tensor with respect to connection
Hence, we can state the following theorem;
Theorem 6 In an LPSasakian manifold Concircular Curvature tensor with respect to Riemannian connection is equal to the Concircular Curvature tensor with respect to Quartersymmetric nonmetric Connection if equation
7.4 holds.8 QuasiConcircular Curvature tensor on an LPSasakian manifold with respect to the Quartersymmetric nonmetric Connection
( , ) ( , )
,Hence, we can state the following theorem;
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